Shimmering Silences in Beautiful Music

An inquiry into the nature of personal experience

A Beauty of MusicB Inquiring into the Experience of MusicC Phases in a Performance of Concert MusicD The Psychology of ResemblancesE Device Models: An Ear for Pythagorean Harmonics and An Ear for Musical Meter

There have been further developments of materials in this website.
Please visit An Ear for Pythagorean Harmonics: Brain Models Built From Timing Devices (12/28/2009), a website and full technical presentation (.pdf file - 460 kB), which have grown out of the materials on this website and which offers advanced developments in timing device systems.

"Ears for Pythagorean Harmonics Used to Investigate
a Procrustean Group of Harmonies (2010)" is the most recent development. (link). A series of advanced device designs begins with the original Ear for Pythagorean harmonics and proceeds to sharpens its perceptions, to augment and extend its range, to give it voice, to join it with similar devices and to use the resulting assembly to investigate the mathematics of harmonic structure.

A. Beauty of Music

A person can experience harmonic beauty while listening to a musical performance. This site inquires into beauty experienced while listening to extracts from a performance of Duo Seraphim, one piece in Claudio Monteverdi's larger work, Vespers of 1610. The extracts have been chosen because they clearly illustrate principles and forms I have developed. The inquiry extends to other kinds of experience, e.g. to rhythmic experience; and the inquiries extend further, into non-musical experience.

B. Inquiring into the Experience of Music

I experience beauty while listening to the Monteverdi extracts. I presume that other persons with an appropriate musical background will experience "much the same" beauty as I do, a presumption supported by the fact that musicians can perform together and achieve a beautiful result. More generally, I presume that experiences of musical beauty are shared in common among musical persons. I presume that beautiful experiences that we share in common arise out of brain activities that we share in common. This inquiry asks: "what is the nature of brain activities that generate such an experience of beauty, shared in common by many persons?"

The inquiry returns with an answer: Our brains generate silences from tones heard together. Generalizing this answer, our brains generate silences from further combinations and sequences of tones.

The word "silence" has an expanded meaning here. "Silences" refers both to a family of signals in brain models and also to features of musical experience. An absence of sound is the "lowest silence" in the family but there are also other family members. Compared to the family of natural numbers (0,1,2,...), an absence of sound resembles "0," the "lowest natural number." An octave is the "next lowest silence," resembling the natural number 1. That is, when two tones are heard together and their relationship is that of the "octave," the brain models will generate a signal that resembles the signal generated when there is an absence of sound, but with a difference that identifies the octave. More generally, each harmony is a silence. Conversely, it is the silences that distinguish harmony from noise.

I am suggesting that we experience both silences and tones; but that we experience a silence in a different way from a tone. There is a single external kind of silence but there are a multitude of internal silences. Often, harmonic relationships are sustained while tones change, e.g., during a chord progression. Each harmonic interval generates a silence and the cluster of harmonics generate a cluster of silences.

The concept of silences is broader than harmonics. A steady meter and rhythm generate a silence; a repeated melodic phrase generates a silence. Each silence has its own character. The silences are like keys of an organ that are played along with the music -- and the beauty is in the silences as well as the tones. The tones are stimulating; but it is in the silences that we find satisfaction and peace.

The inquiry uses two working models to explore the experience of beauty in music and to establish multiple contexts in which silence is an operating principle. "Working" means that they are described by their activities. The models work together in "Phases in a Performance of Concert Music" (part C).

1. The psychological model, the "Psychology of Resemblances" (part D), views music as an activity in the nature of a construction where the raw materials are units of experience that are repeated and varied. An actual construction is carried out by a composer and/or performer who produces, combines and/or sequences units such as musical tones. In the background is the constructivist psychology of pioneering child psychologist Jean Piaget (1896-1980). {Please view my web page on "Jean Piaget and the hazards of the a priori" [...].}

Musical Atoms. Leonard Bernstein (1918-1990) gave Young People's Concerts, with lectures, including one titled "Musical Atoms: A Study of Intervals," where he declared that "those musical protons and electrons (the separate notes, that is) have combined together, forming atoms [the intervals -- octave, second, fourth, etc.] which have then combined into molecules [phrases], which have finally combined into recognizable matter..." An example was the 4th movement of the 4th Symphony of "Brahms, the master builder," who "built almost the whole movement out of the interval of the Third, and its inversion, the interval of the Sixth." Such "atom-molecule" imagery is useful for many constructional purposes and is easy to state and understand.

Bernstein's The Joy of Music has the script for a 1954 telecast during which Bernstein performed Beethoven's rejected sketches for the first movement of the Fifth Symphony and showed why the final version was better: "Beethoven was first and foremost a builder." Compared with the rejected version, "the version that Beethoven finally did use has so much more logic, and builds with so much more ferocity and shock, that there is no comparison." The Psychology of Resemblances seeks to explore and extend the tools of the builders.

The "atom-molecule" approach does not work very well in describing the activities of our brains, including activities of our brains that produce "atom-molecule" constructions. I suggest that our brains work according to different principles that involve resemblances between one complex experience and another complex experience. When two complex voices -- e.g., two human voices -- both sing a tone that has "the same" pitch, the resemblance is "unison" even though the voices are distinct in their spatial locations and in their timbre. "Unison" is a particular kind of resemblance, not an identity. Different resemblances are named "octave," "dominant," etc.

In other words, I suggest that there are brain parts that generate specific resemblances when two complex tones are heard (e.g., two human voices). Each resemblance so generated fits into a structure (or range) of resemblances that, harmonically, runs from consonance to dissonance. Each resemblance also fits into a structure of silences that include harmonic silences, metrical silences and phrasal silences.

Some musical compositions can be organized according to atomic-molecular principles; but our brains, which are generating the experiences, operate according to different principles. We are unconscious of the activity of our brains while we are producing or listening to the music. There is, however, an indirect way of inquiring into unconscious activity and that is the way of device models.

2. Device models are based on technology I have developed for constructing models of brains, as in "an Ear for Pythagorean harmonics." (part E.) In sum, the devices are intended to mimic essential activity of brain parts. My models are entriely different from computer models. I reject principles of mechanistic psychology (e.g., "stimulus-response") and principles of computational neuroscience. (See Links to further materials.)

The simple device model for "follower of the beat" (please see Image below) shows some principles in an elementary way. In the model, the green "beat generator" mimics activity of part of a person's brain. I suggest that what we feel as a beat is produced by the equivalent of a beat generator in our brains. I call this beat the "internal beat." An output from such a beat generator can also control muscular activity, e.g., tapping a foot. The person's physical ear detects sound, what I call an "external beat," but the person's experience does not arise from the external beat. Rather, the external beat has a secondary role.

We begin with a situation where there is a steady beat that is "the same" externally and internally. Operationally, this means that the "beats" on the two sides of the "balancing unit" are equal and that no signal emerges on either the "accelerator" line or the "brake" line. There is a condition of silence in both of these lines. I suggest that these operational features correspond, in fact, to a musical experience where the musician is "in the beat." What the musician hears with ears coincides with what is felt inside: there are two beats that are identical and what is felt is the identity. I suggest that an "internal silence," felt by the person, descrbes this experience. I call the experience "dwelling in the beat" and I point to silence as its operating principle.

A change in the external beat means that different signals go into the two sides of the blue "balancing unit" and this leads to output from that unit and activity in one of the lines. If the internal beat is faster than the external beat, activity on the "brake" line will slow the internal beat generator. If the internal beat is slower than the external beat, activity on the "accelerator" line will speed up the internal beat generator. In each case, activity on a line modifies the follower's internal beat generator to match the external beat and to return the lines to silence. Maintaining as much silence as possible in the lines is "following the beat."

["Dwelling" is a foundational principle in my psychological model of experience. It is self-sustaining activity of the simplest kind. More complex levels of activity are "following" and "balancing." Experiences of "dwelling" also occur while a person is "coasting" in an automobile (no acceleration, no brake) or "gliding" in an airplane.

The "follower of the light" in the adjacent Image is a design for a "bug" that lives in a watery domain also inhabited by "the light." The light-follower uses principles (similar to those operating in the beat-follower) to "follow the light" as the light moves in the domain. The "dwelling" activity of the light-follower is to move "straight ahead toward the light" with both eye sensors equally illuminated and both "motor units" (muscles, governed by modulators) operating at full speed. During "dwelling," the muscle modulators are silent.

If the light shifts away from "straight ahead," one eye receives less illumination than the other, and pulses that are put out by the balancing unit slow down one motor unit. The differential in motor units works physically (mechanically) so that the organism turns and, therefore, "follows the light." The outline of activity is "the same" as for the follower of the beat: a deviation from balance and silence leads to "following" activity that corrects the deviation and restores balance and silence.]

Silence has become an operating principle. The principle is that "controlled operations" means establishing silence as a continuing condition of the lines and directing operations towards a restoration of silence if silence is interrupted. The controlling power of silence is further developed through devices in which silence can trigger signals and signals can trigger silence, as shown in "an Ear for Pythagorean harmonics." (part E).

Things become more interesting if output from a person's internal beat generator is used to drive the person's muscles that strike a drum and produce the external beat. "Dwelling in the beat" continues to mean silence on the accelerator and brake lines. However, now it is also possible for the person to accelerate or brake the beat on the strength of internal influences ("other inputs" in the Image) that may originate in other parts of the person's brain and mind, e.g., those connected to emotions.

For example, erotic excitement leads to acceleration of the beat in the love duet from Act II of Tristan und Isolde, at least until the beat is broken when the lovers are discovered at the climax of their passion -- and a slower pace resumes thereafter. I suggest that, while the conductor is accelerating the beat, the part of the conductor's brain that acts like the balancing unit is producing output on the brake line, but this influence is overpowered by other influences. The conductor's psychological drama tracks the music drama on the stage, where Isolde's servant is vainly attempting to warn her of danger -- thus contributing to the gesamtkunstwerk ("total-artwork"), according to the composer's intention.

As shown in "an Ear for Pythagorean harmonics," the foregoing principles also apply to harmony, where the notion of "different beats" becomes that of "different pitches." The major development is that follower-of-the-beat hears one thing -- it's the same beat or it isn't and if it isn't the same beat, it's so much faster or so much slower; but the Pythagorean ear hears a large number of things. Each thing heard by the ear is a "ratio" and the ear hears ratios through activities of operating devices. Further developments might allow a organism possessing both the ear and instrumental muscles to modify a pitch to establish more perfect silence. Dwelling is activity in which silence is perfectly established.

In other words, a "ratio" is a kind of activity rather than just a number. Each constructed ratio corresponds to a harmony -- or to a dissonance. The ratios fit into a series that runs from consonance to dissonance. Sharp dissonance is most exciting and unison is most relaxing.

The principles also apply to musical meter and there are further applications to other musical elements in a similar fashion but with modifications to fit particular complexities.

C. Phases in a Performance of Concert Music

The Phases (listed above) will be linked to separate pages, as with "Performers." The Phases outline a certain kind of musical performance, like the "phases of the moon" or "stages in a person's life" outline the events described. For example, the device design "follower of the beat" (shown above) is discussed in section 3 ("Following").

Although there are technical statements in the presentation, I hope to make the ideas accessible to anyone interested in music, with just a few technical images.

D. The Psychology of Resemblances

The psychology of resemblances is based on my technical brain models but it stands independently of technology and you can judge its validity based on your own experience. I do not seek to describe all experience but only some experience. The psychology of resemblances provides tools to explore some kinds of experience that are at the "edge of awareness," centered on experiences of silence.

The foundational principles are:

Each of my experiences arises during processes that establish activity patterns in my brain. As a chief mode of operations, a brain generates families of activity patterns that are related by resemblances. When a person acts, some activity patterns are selected and others are extinguished or rejcted. Sometimes a person freely selects among activity patterns to decide the person's next act; and, often, the selection is forced. We know and describe much of our experience, and especially our selections, in terms of resemblances and comparisons. "Atom-molecule" constructions (noted above in B.1) can only partially model such resemblances and comparisons, although some models may be quite good under some limited circumstances.

For musical experience, the principles assert that each sustained element of musical experience corresponds to an activity pattern that is shared among brains, meaning by "element" any of the usual components of music, e.g., meter/rhythm, harmony and melos. Different meters and rhythms are related by resemblances; different harmonies are related by resemblances; and different melodies are related by resemblances.

The families of resmblances and their members are constructional elements of experience, such as elements of musical experience.
"Resemblances" includes "opposites," e.g., melodic and harmonic inversions. The "familes" mentioned in the foundational principles include up-and-down, postive-and-negative, left-and-right. A family may have two members (black-and-white) or an infinite number (rainbow). Resemblances extend across different kinds of experience, enabling us to compare music with motion, music with drama, music with light, even music with architecture. We can see a universe in a grain of sand or a drop of water. According to legend, at least, Newton saw a universal force that acted "the same" on a falling apple and on our Moon circling the Earth.

"Shimmering" occurs when two or more distinct activity patterns are competing for possession of a person's brain. The activity pattern that wins the competition will take the person on the next step of the person's current activity. Sometimes when shimmering occurs, e.g., when a person is selecting from a dinner menu at a restaurant, the person freely decides which activity pattern wins. In composed music, the composer decides for the performers and the audience.

[The central operating principle of my Quad Nets Model of brains is "Shimmering Sensitivity," mimicking a selection of a winning activity pattern in the brain as a result of shifts among and between multiple influences, including very tiny influences that may become all-important in the selection.]

In harmonic music, a harmony is a single, sustained activity pattern - a structure of tones and silences. While things are "in harmony," there is little if any impulse towards movement. Dissonance, on the other hand, creates a shimmering activity pattern, namely, activity of the brain where the next step could take place in more than one direction.

Musical dissonance is a source of "pressure" or "driving force" that became dominant in the 18th and 19th centuries, e.g., in the music of Beethoven and Wagner. Pressure and driving force are found in dramatic situations where the emotional action is "shimmering" and, at least to the persons "in the roles," events can go "either way." Musical force and pressure originated with operatic dramatic performances and developed afterwards into non-vocal dramatic performaces.

Dramatic pressure and force were decidedly different from the steady stream of sound in Renaissance concert music (church and court) prior to 1600. After 1600, Western music was completely transformed, changing from steadiness to drama. Monteverdi's Duo Seraphim was one of the first fruits of that transformation.

In measures 10 through 14 of Extract 1 (see corresponding portion of performing score in the image below), Monteverdi first creates a dissonance and then constructs a path to resolve it that has several steps, each step made up of a a resolution followed by a new dissonance. There is a series of "suspended disonnances." Steps closely resemble other in a repetitious way. In this passage, I hear beauty and repose. I suggest that repeated steps of dissonance and resolution in Extract 1 create a beauty that was novel with Monteverdi and his contemporaries. I suggest that Moteverdi's repetition of steps generates another kind of silence, one that was new in his time. The silences in my brain models suggest an account for this development. Further extensions of the models show how beautiful shimmering silences might be generated through time-displaced phrasal repetitions, e.g., those in Extract 2.

The Phases in a Performance of Concert Music (part C) also participate in the Psychology of Resemblances. According to the Psychology of Resemblances, controlling silences in brain models during an experience of music lead to general operating principles of brains and of minds that guide other kinds of activity. The Phases in a Performance of Concert Music also apply to:

Performances of Muscular Coordination

Please visit "Dancer -- Shimmering Gaits of a Six-legged Engineered Organism." A "gait" is repeated coordinated activity of legs and "shimmering gaits" are switches between gaits that the organism employs, e.g., to climb over an obstacle.

Performances of Verbal Communication

E.g., a lawyer obtaining testimony from a witness by examination in court.

Performances of Dailylife Maintenance

E.g., scheduling and completing errands in a world of detours and delays; making a casserole from a recipe when you don't have all the ingredients.

Performances of Knowledge Construction

E.g., knowledge constructed during the 19th century "paradigm shift" from "the particle theory of light" to "the wave theory of light" as discussed in my Testimony of Freedom.

Performances of Institutional Responsibilities

E.g., resolving a dispute through civil litigation. (Please see material in my archive, The Crucible. Additional materials based on pending cases will be referenced later.)

E. Device Models: An Ear for Phythagorean Harmonics and an Ear for Musical Meter

Device models also participate in the psychology of resemblances. Constructional principles are employed in device designs, as they are in music (see part B). Designs are developed according to "theme and variations."

Device designs come in two large familes, (1) Quad Nets and (2) timing devices. Timing devices have their origin as a specialized kind of Quad Net elemental device (see Image 49.b in the Quad Nets paper), but timing devices have had a separate and fruitful development path. Individual activities of timing devices resemble collective activities of Quad Net device parts. Where a timing device deals with pulses, a Quad Net device part deals with pulse bundles. Pulse bundles come with variations, allowing for selection among variations and for psychological meaning or consequence based on selection.

A technical presentation of "an Ear for Pythagorean Harmonics: Mathematical Processing in Brain-Like Systems of Timing Devices"
(.pdf file, about 223 kB) is available. The principle of "silence" explored in this paper serves as a point of origin for a family of resemblances that starts with harmonic silences and extends through rhythmic and metrical silences to "silences" that guide coordinated muscular action and even more complex psychological activity. (The "Mathematical Processing" paper has been superseded by later developments, as noted elsewhere on this page.)

The technology is entirely conceptual, more in the nature of mathematics than laboratory engineering. None of the designs is in physical existence. Everything is novel. Although there are reference to existing technology, no technical knowledge is required; but you will need a technical aptitude and some sort of technical experience will be helpful.

Technical designs on this web page have been superseded.
Please visit An Ear for Pythagorean Harmonics: Brain Models Built From Timing Devices (12/28/2009), a website that discusses the full technical presentation (.pdf file - 460 kB), which have grown out of the materials on this website.

Please visit other sites in the "Quad Nets" family::

Timing Devices: the original presentation of the timing devices system. The message is that brains are not computers.Quad Nets: general designs for brain-like devices; timing devices are a special class of devices within the more general class.Embodiment of Freedom: integrated models - based on Quad Nets - of brains and experience, physics and psychology.Testimony of Freedom: re-states prior models and extends the inquiry into social and spiritual matters.

Your comments and suggestions are welcome and may be of assistance. Please write to: